Problem: Let $S = \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $S$. Let $N$ be the sum of all of these differences. Find $N.$
When computing $N$, the number $2^x$ will be added $x$ times (for terms $2^x-2^0$, $2^x-2^1$, $\dots,$ $2^x - 2^{x-1}$), and subtracted $10-x$ times. Hence, $N$ can be computed as $$N=10\cdot 2^{10} + 8\cdot 2^9 + 6\cdot 2^8 + \cdots - 8\cdot 2^1 - 10\cdot 2^0.$$Then
\begin{align*}
N & = 10(2^{10}-1) + 8(2^9 - 2^1) + 6(2^8-2^2) + 4(2^7-2^3) + 2(2^6-2^4) \\
& = 10(1023) + 8(510) + 6(252) + 4(120) + 2(48) \\
& = 10(1000+23) + 8(500+10) + 6(250+2) + 480 + 96 \\
&= \boxed{16398}.
\end{align*}